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Open Access
February 28, 2024
Abstract
In this article, we propose an adaptive mesh-refining based on the multi-level algorithm and derive a unified a posteriori error estimate for a class of nonlinear problems. We have shown that the multi-level algorithm on adaptive meshes retains quadratic convergence of Newton’s method across different mesh levels, which is numerically validated. Our framework facilitates to use the general theory established for a linear problem associated with given nonlinear equations. In particular, existing a posteriori error estimates for the linear problem can be utilized to find reliable error estimators for the given nonlinear problem. As applications of our theory, we consider the pseudostress-velocity formulation of Navier–Stokes equations and the standard Galerkin formulation of semilinear elliptic equations. Reliable and efficient a posteriori error estimators for both approximations are derived. Finally, several numerical examples are presented to test the performance of the algorithm and validity of the theory developed.
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February 21, 2024
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We derive and analyze a symmetric interior penalty discontinuous Galerkin scheme for the approximation of the second-order form of the radiative transfer equation in slab geometry. Using appropriate trace lemmas, the analysis can be carried out as for more standard elliptic problems. Supporting examples show the accuracy and stability of the method also numerically, for different polynomial degrees. For discretization, we employ quad-tree grids, which allow for local refinement in phase-space, and we show exemplary that adaptive methods can efficiently approximate discontinuous solutions. We investigate the behavior of hierarchical error estimators and error estimators based on local averaging.
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February 10, 2024
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This article discusses the quasi-optimality of adaptive nonconforming finite element methods for distributed optimal control problems governed by 𝑚-harmonic operators for m = 1 , 2 m=1,2 . A variational discretization approach is employed and the state and adjoint variables are discretized using nonconforming finite elements. Error equivalence results at the continuous and discrete levels lead to a priori and a posteriori error estimates for the optimal control problem. The general axiomatic framework that includes stability, reduction, discrete reliability, and quasi-orthogonality establishes the quasi-optimality. Numerical results demonstrate the theoretically predicted orders of convergence and the efficiency of the adaptive estimator.
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January 31, 2024
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This paper aims to investigate the weak convergence of the Rosenbrock semi-implicit method for semilinear parabolic stochastic partial differential equations (SPDEs) driven by additive noise. We are interested in SPDEs where the nonlinear part is stronger than the linear part, also called stochastic reaction dominated transport equations. For such SPDEs, many standard numerical schemes lose their stability properties. Exponential Rosenbrock and Rosenbrock-type methods were proved to be efficient for such SPDEs, but only their strong convergence were recently analyzed. Here, we investigate the weak convergence of the Rosenbrock semi-implicit method. We obtain a weak convergence rate which is twice the rate of the strong convergence. Our error analysis does not rely on Malliavin calculus, but rather only uses the Kolmogorov equation and the smoothing properties of the resolvent operator resulting from the Rosenbrock semi-implicit approximation.
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January 30, 2024
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This paper develops and analyses a semi-discrete numerical method for the two-dimensional Vlasov–Stokes system with periodic boundary condition. The method is based on the coupling of the semi-discrete discontinuous Galerkin method for the Vlasov equation with discontinuous Galerkin scheme for the stationary incompressible Stokes equation. The proposed method is both mass and momentum conservative. Since it is difficult to establish non-negativity of the discrete local density, the generalized discrete Stokes operator become non-coercive and indefinite, and under the smallness condition on the discretization parameter, optimal error estimates are established with help of a modified the Stokes projection to deal with the Stokes part and, with the help of a special projection, to tackle the Vlasov part. Finally, numerical experiments based on the dG method combined with a splitting algorithm are performed.
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January 30, 2024
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We consider discontinuous Galerkin methods for an elliptic distributed optimal control problem, and we propose multigrid methods to solve the discretized system. We prove that the 𝑊-cycle algorithm is uniformly convergent in the energy norm and is robust with respect to a regularization parameter on convex domains. Numerical results are shown for both 𝑊-cycle and 𝑉-cycle algorithms.
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January 25, 2024
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In this paper, we consider the inverse problem of vibro-acoustography, a technique for enhancing ultrasound imaging by making use of nonlinear effects. It amounts to determining two spatially variable coefficients in a system of PDEs describing propagation of two directed sound beams and the wave resulting from their nonlinear interaction. To justify the use of Newton’s method for solving this inverse problem, on one hand, we verify well-definedness and differentiability of the forward operator corresponding to two versions of the PDE model; on the other hand, we consider an all-at-once formulation of the inverse problem and prove convergence of Newton’s method for its solution.
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January 25, 2024
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This paper deals with the efficient implementation of the finite element method with continuous piecewise linear functions (P1-FEM) in R d \mathbb{R}^{d} ( d ∈ N d\in\mathbb{N} ). Although at present there does not seem to be a very high practical demand for finite element methods that use higher-dimensional simplicial partitions, there are some advantages in studying the efficient implementation of the method independent of the dimension. For instance, it provides additional insights into necessary data structures and the complexity of implementations. Throughout, the focus is on an efficient realization using Matlab built-in functions and vectorization. The fast and vectorized Matlab function can be easily implemented in many other vector languages and is provided in Julia, too. The complete implementation of the adaptive FEM is given, including assembling stiffness matrix, building load vector, error estimation, and adaptive mesh-refinement. Numerical experiments underline the efficiency of our freely available code which is observed to be of a slightly more than linear complexity with respect to the number of elements when memory limits are not exceeded.
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January 25, 2024
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We present a recently developed preconditioning of square block matrices (PRESB) to be used within a parallel method to solve linear systems of equations arising from tensor-product discretizations of initial boundary-value problems for parabolic partial differential equations. We consider weak formulations in Bochner–Sobolev spaces and tensor-product finite element approximations for the heat and eddy current equations. The fast diagonalization method is employed to decouple the arising linear system of equations into auxiliary spatial complex-valued linear systems that can be solved concurrently. We prove that the real part of the system matrix is positive definite, which allows us to accelerate the flexible generalized minimal residual method (FGMRES) by the PRESB preconditioner. The action of PRESB on a vector includes two solutions with positive definite matrices. The spectrum of the preconditioned system lies between 1/2 and 1. Finally, we combine the PRESB-FGMRES method with multigrid-CG iterations and illustrate the numerical efficiency and the robustness for spatial discretizations up to 12 millions degrees of freedom.
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January 25, 2024
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We compare different machine learning estimators and present details about their implementation in Python. The computational studies are conducted for classification as well as regression problems. Moreover, as one of the founding problems of machine learning, we present the specific classification task of handwritten digit recognition . In this connection, we discuss the mathematical formulation and of course the implementation details of this problem. All corresponding Python code is fully provided on request and can be downloaded from the author’s GitHub page https://github.com/Fab1Fatal.
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January 6, 2024
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In this article, convergence and quasi-optimal rate of convergence of an adaptive finite element method (in short, AFEM) is shown for a general second-order non-selfadjoint elliptic PDE with convection term b ∈ [ L ∞ ( Ω ) ] d {b\in[L^{\infty}(\Omega)]^{d}} and using minimal regularity of the dual problem, i.e., the solution of the dual problem has only H 1 {H^{1}} -regularity, which extends the result [J. M. Cascon, C. Kreuzer, R. H. Nochetto and K. G. Siebert, Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. Anal. 46 2008, 5, 2524–2550]. The theoretical results are illustrated by numerical experiments.
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January 4, 2024
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We consider the numerical approximation of Gaussian random fields on closed surfaces defined as the solution to a fractional stochastic partial differential equation (SPDE) with additive white noise. The SPDE involves two parameters controlling the smoothness and the correlation length of the Gaussian random field. The proposed numerical method relies on the Balakrishnan integral representation of the solution and does not require the approximation of eigenpairs. Rather, it consists of a sinc quadrature coupled with a standard surface finite element method. We provide a complete error analysis of the method and illustrate its performances in several numerical experiments.
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January 2, 2024
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The problem of optimal recovering high-order mixed derivatives of bivariate functions with finite smoothness is studied. Based on the truncation method, an algorithm for numerical differentiation is constructed, which is order-optimal both in the sense of accuracy and in terms of the amount of involved Galerkin information. Numerical examples are provided to illustrate the fact that our approach can be implemented successfully.
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January 2, 2024
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This work is concerned with the classical wave equation with a high-contrast coefficient in the spatial derivative operator. We first treat the periodic case, where we derive a new limit in the one-dimensional case. The behavior is illustrated numerically and contrasted to the higher-dimensional case. For general unstructured high-contrast coefficients, we present the Localized Orthogonal Decomposition and show a priori error estimates in suitably weighted norms. Numerical experiments illustrate the convergence rates in various settings.
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Open Access
January 2, 2024
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The JPEG algorithm is a defacto standard for image compression. We investigate whether adaptive mesh refinement can be used to optimize the compression ratio and propose a new adaptive image compression algorithm. We prove that it produces a quasi-optimal subdivision grid for a given error norm with high probability. This subdivision can be stored with very little overhead and thus leads to an efficient compression algorithm. We demonstrate experimentally, that the new algorithm can achieve better compression ratios than standard JPEG compression with no visible loss of quality on many images. The mathematical core of this work shows that Binev’s optimal tree approximation algorithm is applicable to image compression with high probability, when we assume small additive Gaussian noise on the pixels of the image.
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November 28, 2023
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This work presents the multiharmonic analysis and derivation of functional type a posteriori estimates of a distributed eddy current optimal control problem and its state equation in a time-periodic setting. The existence and uniqueness of the solution of a weak space-time variational formulation for the optimality system and the forward problem are proved by deriving inf-sup and sup-sup conditions. Using the inf-sup and sup-sup conditions, we derive guaranteed, sharp and fully computable bounds of the approximation error for the optimal control problem and the forward problem in the functional type a posteriori estimation framework. We present here the first computational results on the derived estimates.
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November 25, 2023
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In this paper, we apply a two-grid scheme to the DG formulation of the 2D transient Navier–Stokes model. The two-grid algorithm consists of the following steps: Step 1 involves solving the nonlinear system on a coarse mesh with mesh size 𝐻, and Step 2 involves linearizing the nonlinear system by using the coarse grid solution on a fine mesh of mesh size ℎ and solving the resulting system to produce an approximate solution with desired accuracy. We establish optimal error estimates of the two-grid DG approximations for the velocity and pressure in energy and L 2 L^{2} -norms, respectively, for an appropriate choice of coarse and fine mesh parameters. We further discretize the two-grid DG model in time, using the backward Euler method, and derive the fully discrete error estimates. Finally, numerical results are presented to confirm the efficiency of the proposed scheme.
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November 23, 2023
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We construct a C 1 C^{1} - P 7 P_{7} Bell finite element by restricting its normal derivative from a P 6 P_{6} polynomial to a P 5 P_{5} polynomial, and its second normal derivative from a P 5 P_{5} polynomial to a P 4 P_{4} polynomial, on the three edges of every triangle. On one triangle, the finite element space contains the P 6 P_{6} polynomial space. We show the method converges at order 7 in L 2 L^{2} -norm. By eliminating all degrees of freedom on edges of C 1 C^{1} - P 7 P_{7} Argyris finite element, the global degrees of freedom of the new element are reduced substantially from 27 V 27V to 12 V 12V asymptotically, where 𝑉 is the number of vertices in the triangular mesh. While the global degrees of freedom of the C 1 C^{1} - P 6 P_{6} Argyris finite element is 19 V 19V , the new element is equally accurate but more economic. Numerical tests are presented, showing the new element is more accurate than the existing element while having less global unknowns.
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November 21, 2023
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Reconstructing the pressure from given flow velocities is a task arising in various applications, and the standard approach uses the Navier–Stokes equations to derive a Poisson problem for the pressure p . That method, however, artificially increases the regularity requirements on both solution and data. In this context, we propose and analyze two alternative techniques to determine p ∈ L 2 ( Ω ) {p\in L^{2}(\Omega)} . The first is an ultra-weak variational formulation applying integration by parts to shift all derivatives to the test functions. We present conforming finite element discretizations and prove optimal convergence of the resulting Galerkin–Petrov method. The second approach is a least-squares method for the original gradient equation, reformulated and solved as an artificial Stokes system. To simplify the incorporation of the given velocity within the right-hand side, we assume in the derivations that the velocity field is solenoidal. Yet this assumption is not restrictive, as we can use non-divergence-free approximations and even compressible velocities. Numerical experiments confirm the optimal a priori error estimates for both methods considered.
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Open Access
November 1, 2023
Abstract
We present an adaptive absorbing boundary layer technique for the nonlinear Schrödinger equation that is used in combination with the Time-splitting Fourier spectral method (TSSP) as the discretization for the NLS equations. We propose a new complex absorbing potential (CAP) function based on high order polynomials, with the major improvement that an explicit formula for the coefficients in the potential function is employed for adaptive parameter selection. This formula is obtained by an extension of the analysis in [R. Kosloff and D. Kosloff, Absorbing boundaries for wave propagation problems, J. Comput. Phys. 63 1986, 2, 363–376]. We also show that our imaginary potential function is more efficient than what is used in the literature. Numerical examples show that our ansatz is significantly better than existing approaches. We show that our approach can very accurately compute the solutions of the NLS equations in one dimension, including in the case of multi-dominant wave number solutions.
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October 11, 2023
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This article develops and analyses a conforming virtual element scheme for the spatial discretization of parabolic integro-differential equations combined with backward Euler’s scheme for temporal discretization. With the help of Ritz–Voltera and L 2 L^{2} projection operators, optimal a priori error estimates are established. Moreover, several numerical experiments are presented to confirm the computational efficiency of the proposed scheme and validate the theoretical findings.
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October 6, 2023
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We analyze a numerical method to solve the time-dependent linear Pauli equation in three space dimensions. The Pauli equation is a semi-relativistic generalization of the Schrödinger equation for 2-spinors which accounts both for magnetic fields and for spin, with the latter missing in preceding numerical work on the linear magnetic Schrödinger equation. We use a four term operator splitting in time, prove stability and convergence of the method and derive error estimates as well as meshing strategies for the case of given time-independent electromagnetic potentials, thus providing a generalization of previous results for the magnetic Schrödinger equation.
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October 5, 2023
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We prove the convergence of an incremental projection numerical scheme for the time-dependent incompressible Navier–Stokes equations, without any regularity assumption on the weak solution. The velocity and the pressure are discretized in conforming spaces, whose compatibility is ensured by the existence of an interpolator for regular functions which preserves approximate divergence-free properties. Owing to a priori estimates, we get the existence and uniqueness of the discrete approximation. Compactness properties are then proved, relying on a Lions-like lemma for time translate estimates. It is then possible to show the convergence of the approximate solution to a weak solution of the problem. The construction of the interpolator is detailed in the case of the lowest degree Taylor–Hood finite element.
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October 4, 2023
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We construct three H-curl-curl finite elements. The P 2 P_{2} and P 3 P_{3} vector finite element spaces are both enriched by one common P 4 P_{4} bubble and their local degrees of freedom are 13 and 21, respectively. As there does not exist any P 1 P_{1} H-curl-curl conforming finite element, the P 1 P_{1} H-curl-curl nonconforming finite element is constructed with three additional P 4 P_{4} bubbles. Numerical tests are presented, confirming the conformity and the optimal order of convergence.
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October 4, 2023
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A high-performance sparse model is very important for processing high-dimensional data. Therefore, based on the quadratic approximate greed pursuit (QAGP) method, we can make full use of the information of the quadratic lower bound of its approximate function to get the relaxation quadratic approximate greed pursuit (RQAGP) method. The calculation process of the RQAGP method is to construct two inexact quadratic approximation functions by using the m -strongly convex and L -smooth characteristics of the objective function and then solve the approximation function iteratively by using the Iterative Hard Thresholding (IHT) method to get the solution of the problem. The convergence analysis is given, and the performance of the method in the sparse logistic regression model is verified on synthetic data and real data sets. The results show that the RQAGP method is effective.
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August 25, 2023
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This paper is devoted to the study of Bingham flow with variable density. We propose a local bi-viscosity regularization of the stress tensor based on a Huber smoothing step. Next, our computational approach is based on a second-order, divergence-conforming discretization of the Huber regularized Bingham constitutive equations, coupled with a discontinuous Galerkin scheme for the mass density. We take advantage of the properties of divergence-conforming and discontinuous Galerkin formulations to effectively incorporate upwind discretizations, thereby ensuring the stability of the formulation. The stability of the continuous problem and the fully discrete scheme are analyzed. Further, a semismooth Newton method is proposed for solving the obtained fully discretized system of equations at each time step. Finally, several numerical examples that illustrate the main features of the problem and the properties of the numerical scheme are presented.
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Open Access
August 17, 2023
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We present the self-consistent Pauli equation, a semi-relativistic model for charged spin- 1 / 2 1/2 particles with self-interaction with the electromagnetic field. The Pauli equation arises as the O ( 1 / c ) O(1/c) approximation of the relativistic Dirac equation. The fully relativistic self-consistent model is the Dirac–Maxwell equation where the description of spin and the magnetic field arises naturally. In the non-relativistic setting, the correct self-consistent equation is the Schrödinger–Poisson equation which does not describe spin and the magnetic field and where the self-interaction is with the electric field only. The Schrödinger–Poisson equation also arises as the mean field limit of the 𝑁-body Schrödinger equation with Coulomb interaction. We propose that the Pauli–Poisson equation arises as the mean field limit N → ∞ N\to\infty of the linear 𝑁-body Pauli equation with Coulomb interaction where one has to pay extra attention to the fermionic nature of the Pauli equation. We present the semiclassical limit of the Pauli–Poisson equation by the Wigner method to the Vlasov equation with Lorentz force coupled to the Poisson equation which is also consistent with the hierarchy in 1 / c 1/c of the self-consistent Vlasov equation. This is a non-trivial extension of the groundbreaking works by Lions & Paul and Markowich & Mauser, where we need methods like magnetic Lieb–Thirring estimates.
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August 8, 2023
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A second-order backward differentiation formula (BDF2) finite-volume discretization for a nonlinear cross-diffusion system arising in population dynamics is studied. The numerical scheme preserves the Rao entropy structure and conserves the mass. The existence and uniqueness of discrete solutions and their large-time behavior as well as the convergence of the scheme are proved. The proofs are based on the G-stability of the BDF2 scheme, which provides an inequality for the quadratic Rao entropy and hence suitable a priori estimates. The novelty is the extension of this inequality to the system case. Some numerical experiments in one and two space dimensions underline the theoretical results.
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August 8, 2023
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In this paper, some symmetrized two-scale finite element methods are proposed for a class of partial differential equations with symmetric solutions. With these methods, the finite element approximation on a fine tensor-product grid is reduced to the finite element approximations on a much coarser grid and a univariant fine grid. It is shown by both theory and numerics including electronic structure calculations that the resulting approximations still maintain an asymptotically optimal accuracy. By symmetrized two-scale finite element methods, the computational cost can be reduced further by a factor of 𝑑 approximately compared with two-scale finite element methods when Ω = ( 0 , 1 ) d \Omega=(0,1)^{d} . Consequently, symmetrized two-scale finite element methods reduce computational cost significantly.
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August 8, 2023
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We consider the 3D stochastic Navier–Stokes equation on the torus. Our main result concerns the temporal and spatio-temporal discretisation of a local strong pathwise solution. We prove optimal convergence rates for the energy error with respect to convergence in probability, that is convergence of order (up to) 1 in space and of order (up to) 1/2 in time. The result holds up to the possible blow-up of the (time-discrete) solution. Our approach is based on discrete stopping times for the (time-discrete) solution.
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July 29, 2023
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Inhomogeneous essential boundary conditions can be appended to a well-posed PDE to lead to a combined variational formulation. The domain of the corresponding operator is a Sobolev space on the domain Ω on which the PDE is posed, whereas the codomain is a Cartesian product of spaces, among them fractional Sobolev spaces of functions on ∂ Ω \partial\Omega . In this paper, easily implementable minimal residual discretizations are constructed which yield quasi-optimal approximation from the employed trial space, in which the evaluation of fractional Sobolev norms is fully avoided.
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July 25, 2023
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We consider an inverse problem for the Helmholtz equation of reconstructing a solution from measurements taken on a segment inside a semi-infinite strip. Homogeneous Neumann conditions are prescribed on both side boundaries of the strip and an unknown Dirichlet condition on the remaining part of the boundary. Additional complexity is that the radiation condition at infinity is unknown. Our aim is to find the unknown function in the Dirichlet boundary condition and the radiation condition. Such problems appear in acoustics to determine acoustical sources and surface vibrations from acoustic field measurements. The problem is split into two sub-problems, a well-posed and an ill-posed problem. We analyse the theoretical properties of both problems; in particular, we show that the radiation condition is described by a stable non-linear problem. The second problem is ill-posed, and we use the Landweber iteration method together with the discrepancy principle to regularize it. Numerical tests show that the approach works well.
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July 25, 2023
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This paper proposes novel computational multiscale methods for linear second-order elliptic partial differential equations in nondivergence form with heterogeneous coefficients satisfying a Cordes condition. The construction follows the methodology of localized orthogonal decomposition (LOD) and provides operator-adapted coarse spaces by solving localized cell problems on a fine scale in the spirit of numerical homogenization. The degrees of freedom of the coarse spaces are related to nonconforming and mixed finite element methods for homogeneous problems. The rigorous error analysis of one exemplary approach shows that the favorable properties of the LOD methodology known from divergence-form PDEs, i.e., its applicability and accuracy beyond scale separation and periodicity, remain valid for problems in nondivergence form.
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July 20, 2023
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This paper defines analysis-suitable T-splines for arbitrary degree (including even and mixed degrees) and arbitrary dimension. We generalize the concept of anchor elements known from the two-dimensional setting, extend existing concepts of analysis-suitability and show their sufficiency for linearly independent T-splines.
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May 9, 2023
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We study variants of the mixed finite element method (mixed FEM) and the first-order system least-squares finite element (FOSLS) for the Poisson problem where we replace the load by a suitable regularization which permits to use H − 1 H^{-1} loads. We prove that any bounded H − 1 H^{-1} projector onto piecewise constants can be used to define the regularization and yields quasi-optimality of the lowest-order mixed FEM resp. FOSLS in weaker norms. Examples for the construction of such projectors are given. One is based on the adjoint of a weighted Clément quasi-interpolator. We prove that this Clément operator has second-order approximation properties. For the modified mixed method, we show optimal convergence rates of a postprocessed solution under minimal regularity assumptions—a result not valid for the lowest-order mixed FEM without regularization. Numerical examples conclude this work.
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March 29, 2023
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For the eigenvalue problem of the Steklov differential operator, an algorithm based on the conforming finite element method (FEM) is proposed to provide guaranteed lower bounds for the eigenvalues. The proposed lower eigenvalue bounds utilize the a priori error estimation for FEM solutions to non-homogeneous Neumann boundary value problems, which is obtained by constructing the hypercircle for the corresponding FEM spaces and boundary conditions. Numerical examples demonstrate the efficiency of our proposed method.